Paper Summaries • Any takers? Material Properties Assignments Projects • Proposals – Should all have received feedback via e-mail. • Checkpoint 2 – Grading – Due Wednesday • ACCEPTED – Any questions? • CONDITIONALLY ACCEPTED • PLEASE RESUBMIT WITH MORE INFO Projects Plan for today • Project feedback • Material Properties – Those who have not sent proposal, please see me! – Bi-directional reflectance distribution functions • Approx 18 projects (BRDFs) • Listing of projects now on Web – Illumination Models • Presentation schedule – Using Empirical Data – Presentations (15 min max) – Last 3 classes (week 10 + finals week) – Sign up • Email me with 1 st , 2 nd , 3 rd choices • First come first served. 1
Shading Computer Graphics as Virtual Photography real camera photo Photographic • Computing the light that leaves a point Photography: scene (captures processing print • Shading point - point under investigation light) • Illumination model - function or algorithm used to describe the reflective characteristics of a given processing surface. • Shading model – algorithm for using an illumination camera Computer 3D synthetic tone model to determine the color of a point on a surface. model Graphics: models image reproduction • For efficiency’s sake, most illumination models are (focuses simulated approximations. lighting) Bi-directional Reflectance Functions (BRDF) Reflections • Ambient – light uniformly incident from the environment • Diffuse – light scattered equally in all directions • Ambient and Diffuse – color of material plays a part • Specular – highlights connected with mirrorness • Specular – mostly color of light BRDF Example BRDF Example Note: Note: Hidden Specular shadows highlights Sun behind observer Sun opposite observer Sun behind observer Sun opposite observer 2
BRDF Example BRDF • Bi-directional Reflectance Function = φ θ φ θ Note: Note: BRDF f ( , , , ) Hidden Specular r i i r r shadows highlights At a given point, gives relative reflected illumination in any direction with respect to incoming illumination coming from any direction; Note: The θ ’s are elevation, ϕ ’s are measured about the surface normal. The i ’s refer to the incident ray; the r ’s to the reflected ray. Sun behind observer Sun opposite observer BRDF BRDF Geometry • Can return any positive value. • Generally wavelength specific. BRDF = φ θ φ θ λ f ( , , , , ) r i i r r Illumination Models BRDF • Illumination model - function or algorithm • Simplifying Assumptions wrt the BRDF used to describe the reflective – Light enters and leaves from the same point. characteristics of a given surface. • Not necessarily true • Subsurface scattering • Revise to… • Skin, marble – function or algorithm used in approximating the – Light of a given wavelength will only reflect back light of BRDF. that same wavelength • Not necessarily true • Light Interference • Oily patches, peacock feathers 3
Illumination Modeling Illumination Models • Three approaches • Illumination Models and Viewing Direction – Heuristic – Generally, BRDFs are independent of viewing • The kludge! direction • Usually simple, yet not physically based – Most Illumination models take viewing – Simulation direction into consideration • Employ physical model • More complex than heuristic, but more accurate – Empirical • Use measured samples Illumination Models Illumination Models • Geometry • Geometry – N - normal vector N V H – S - direction of incoming light viewer normal – R - direction of perfect mirror reflection Half-way – H - halfway between light direction and R S reflection viewing direction. source – V - viewing direction. Illumination Models Illumination Models • Recall from Linear Algebra • Lambertian – Physically based…but limited u • Phong θ – heuristic • Cook-Torrance v – Physically based • = θ • Ward u v u v cos – heuristic based on physics – Ansitropic reflection Just one reason to normalize! • Many others! 4
Illumination Models Lambertian Model • BRDF Viewer • Lambert Model – bv by Szymon Rusinkiewicz (Princeton) – Perfectly diffuse surface – http://graphics.stanford.edu/~smr/brdf/bv – reflection is constant in all directions (k d ) – SGI, Linux, and Java versions although not – Independent of viewer direction readily available for Java. I have it, if you want it, and you’ll need to load Java3D as well! Lambertian Model Lambertian Model • Lambert Model • Lambert Model – why cos θ ? – Surface has differential area dA – Intensity varies with projected area on surface – Projected area = cos θ = θ = • L ( V ) L S k cos L ( V ) L k ( N S ) d S d Lambertian Model Phong Model • BRDF Viewer • Phong Model – introduces specular (mirror-like) reflections http://graphics.stanford.edu/~smr/brdf/bv – Viewer direction becomes more important – three components • ambient - background light (k a ) • diffuse - Lambertian reflection (k d ) • specular – mirror-like reflection(k s ) 5
Phong Model Phong-Blinn Model – Uses halfway angle rather than reflected ∑ ∑ = + • + • k L ( V ) k L k L ( S N) k L ( R V) e a a d i i s i i i i ∑ ∑ = + • + • ambient k diffuse specular L ( V ) k L k L ( S N) k L ( H N) e a a d i i s i i i i ambient diffuse specular Note: L n are radiance terms, include both light and material info Phong-Blinn Model Cook-Torrance Model – based on physics of a surface • BRDF Viewer • Actually developed by Torrance & Sparrow, http://graphics.stanford.edu/~smr/brdf/bv physicists. • Jim Blinn was the first to apply to CG • Cook & Torrance’s was the first complete implementation – components • microfacet model - describes geometry of surface • Fresnel term - describes reflectance • Roughness - describes microfacet distribution. Cook-Torrance Model Cook-Torrance Model • Microfacets • Microfacets – surface is composed of V shaped grooves (microfacets) – Light interactions with microfacets • Reflect - causes specular reflections • Scatter - causes diffuse reflections 6
Cook-Torrance Model Cook-Torrance Model • Microfacets – GeometryTerm • Fresnel Equation for polarized light – Some microfacets may shadow others – Describes reflectance as a function of: • Wavelength of incident light ( λ ) • Index of refraction ( η ( λ )) • Extinction coefficient (ease at which wave can penetrate a surface) ( κ ( λ )) • Angle of incidence ( θ ) ⎧ ⎫ • • • • 2 ( N H)(N V) 2 ( N H)(N S) = ⎨ ⎬ G min 1 , , • • ⎩ ⎭ (V H) (V H) Note: S from before is the L in these diagrams Cook-Torrance Model Cook-Torrance Model • Fresnel equations for polarized light • Fresnel + − θ + θ 2 2 2 a b 2 a cos cos = Perpendicular component – If all quantities known, use Fresnel equations Fs + + θ + θ 2 2 2 2 cos cos a b a – If not, approximate using reflectance off normal • See [Glassner] or [Cook/Torrance81] for details + − θ θ + θ θ 2 2 2 2 a b 2 a sin tan sin tan Parallel component = F F + + θ θ + θ θ p s 2 2 2 2 a b 2 a sin tan sin tan a, b are functions 1 1 = + of η , κ , and θ F F F s p 2 2 η , κ are functions F is total reflectance of λ Cook-Torrance Model Cook-Torrance Model • Roughness • Roughness – Characterizes the distribution of the slopes of the microfacets – Roughness parameter, m • m between 0 -1 • small m - smooth surface, specular reflectance − α = − α 2 2 ((tan ) / m ) ( / m ) e D ce • large m - rough surface, diffuse reflectance = D α 2 4 m cos – Statistical models Gaussian Model Beekman Model c is arbitrary constant 7
Cook-Torrance Model Cook-Torrance Model • Roughness • Putting it all together = + f sf df r s d specular diffuse total reflectance × × 1 1 F D G = = f f s π d π • • ( N S)(N V) Where D is the roughness function, F is the Fresnel function, and G is the geometrical attenuation factor from previous pages Cook-Torrance Model Cook-Torrance Models • Complete Cook-Torrance Model • examples ∑ = + • ϖ L L R L ( N S i )( f ) d r a a i r i i Parameters for f r : � � m – roughness value � Type of material (determines terms for Fresnel eqn) � Wavelength of incident light (determines terms for Fresnel eqn) � Diffuse / specular contribution constants � L a R a is the ambient radiance reflected by R a � L i is the light’s radiance Cook-Torrance Models Cook-Torrance Model • BRDF Viewer • Summary – Complicated model based on physics http://graphics.stanford.edu/~smr/brdf/bv – Components • Microfacets • Fresnel equation • Roughness – Want accuracy? Go to the source! • Break. 8
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